SCHWEITZER 2012 Problem 11

Let $X_1,X_2,..$ be independent random variables with the same distribution, and let $S_n=X_1+X_2+...+X_n, n=1,2,...$ . For what real numbers $c$ is the following statement true: $$P\left(\left| \frac{S_{2n}}{2n}- c \right| \leqslant \left| \frac{S_n}{n}-c\right| \right)\geqslant \frac{1}{2}$$